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Full text of "Men Of Mathematics"

COMPLETE INDEPENDENCE
*I a. If a and b are in the class K, then a + 6 is in the class Km
'I b. // a and b are in the class K, then ab is in the class K.
4II a. There is an element Z such that a + Z = a for every
element a.                                                                               *
'II b. There is an element U such that aU = a for every
element a.
'Ill a. a -f- b = 6 -f a.
'Ill b. ab = ba.
•IV a. a -f be = (a -f 6) (a -f c).
'IV b. a(& + c) = ab + ac.
'V. Tor eeen/ element a there is an element a' such that a -f of
= U and aa' — Z.
4VL There are at least too distinct elements in the class KS
It will be readily seen that these postulates are satisfied by
the following interpretation: a, b, c, ... are classes', a + b is the
class of all those things that are in at least one of the classes,
a, b; ab is the class of all those things that are in both of the
classes a, b; Z is the 'null class' -the class that has no members;
U is the 'universal class' - the class that contains all the things
in all the classes under discussion. Postulate V then states that
given any class a, there is a class a' consisting of all those things
which are not in a. Note that VI implies that U, Z are not the
same class.
From such a simple and obvious set of statements it seems
rather remarkable that the whole of classical logic can be built
up symbolically by means of the easy algebra generated by the
postulates. From these postulates a theory of what may be
called 'logical equations' is developed: problems hi logic are
translated into such equations, which are then 'solved' by the
devices of the algebra; the solution is then reinterpreted in
terms of the logical data, giving the solution of the original
problem. We shall close this description with the symbolic
equivalent of 'inclusion' - also interpretable, when propositions
rather than classes are the elements of K9 as 'implication'.
'The relation a <b [read, a is included in b] is defined by any
one of the following equations
a + b = b, ab = a, a' + b = U, ab' = Z:
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